
The 2D2V low rank algorithm presented in [1]}. In particular, starting from the solution \({\bf f}^n\)  in the low rank format (REF ), we can add basis from a step of time integration (e.g. second order multi-step method) to obtain the intermediate solution \({\bf f}^{n+1, \star }\)  in the same low rank format but with higher rank.

Update macroscopic solutions \(\rho ^{n+1, M}\) , \({\bf J}_1^{n+1, M}\) , \({\bf J}_2^{n+1, M}\) , \(\kappa ^{n+1, M}\)  by using a conservative high order finite difference scheme with KFVS, coupled with the corresponding time integrator, to solve the macroscopic conservation laws (REF )-() in a 2D setting.

Perform the decomposition of \({\bf f}^{n+1, \star }\)  obtained from Step 1 as \({\bf f}^{n+1, \star } = P_{\mathcal {N}} ({\bf f}) + (I - P_{\mathcal {N}})({\bf f}) \doteq {\bf f}_1 + {\bf f}_2\) .

Replace \({\bf f}_1\)  with a new \({\bf f}_1^M\)  via (REF ), but with \(\rho ^{n+1, M}\) , \({\bf J}_1^{n+1, M}\) , \({\bf J}_2^{n+1, M}\)  obtained from Step 2 for local conservation of macroscopic variables.

Perform a weighted SVD truncation to \({\bf f}_2\) , followed with a projection operator \((I-P_{\mathcal {N}})\)  to ensure zero mass momentum and kinetic energy. That is, to compute \((I-P_{\mathcal {N}}) (\sqrt{\bf w}\star \mathcal {T}_\varepsilon (\frac{1}{\sqrt{\bf w}}\star {\bf f}_2))\) .

The updated \({\bf f}^{n+1} = {\bf f}_1^M + (I-P_{\mathcal {N}}) (\sqrt{\bf w} \star \mathcal {T}_\varepsilon (\frac{1}{\sqrt{\bf w}}\star {\bf f}_2))\)  from previous two sub-steps.


